#8096633

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26)(27,36)(28,33)(29,39)(30,34)(31,35)(32,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27)(26,29)(28,37)(30,38)(31,39)(32,40)(33,34)(35,36);(1,3)(2,5)(4,8)(6,7)(25,28)(26,32)(27,30)(29,34)(31,38)(33,35)(36,40)(37,39);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24)(25,29,31,36)(26,27,35,39)(28,34,38,40)(30,33,37,32);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,16,17,18)(12,22,23,19)(14,20,21,24)(25,28)(26,33)(27,37)(29,34)(30,39)(31,38)(32,35)(36,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,30,31,37)(26,34,35,40)(27,38,39,28)(29,33,36,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003829);